SOME SEQUENCE SPACES AND STATISTICAL CONVERGENCE

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SOME SEQUENCE SPACES AND STATISTICAL CONVERGENCE

E. SAVA¸S Received 29 January 1999

We introduce the strongly (V ,λ)-convergent sequences and give the relation between strongly(V ,λ)-convergence and strongly(V ,λ)-convergence with respect to a modulus.

2000 Mathematics Subject Classification: 40D25, 40A05, 40C05.

1. Introduction. Let λ=(λn) be a nondecreasing sequence of positive numbers tending to∞, andλn+1≤λn+1, λ1=1.

The generalized de la Vallée-Poussin mean is defined by tn= 1

λn

k∈In

xk, (1.1)

whereIn=[n−λn+1,n]. A sequencex=(xk)is said to be (V ,λ)-summable to a numberL(see [5]) iftn(x)→Lasn→ ∞. Ifλn=n, then(V ,λ)-summability is reduced to(C,1)-summability. We write

[V ,λ]= x=

xk

: for someL,lim

n

1 λn

k∈In

xk−L=0

(1.2)

for sets of sequencesx=(xk)which are strongly(V ,λ)-summable toL, that is,xk→ L[V ,λ].

We recall that a modulusf is a function from[0,∞)to[0,∞)such that (i) f (x)=0 if and only ifx=0;

(ii) f (x+y)≤f (x)+f (y)for allx, y≥0;

(iii) f is increasing;

(iv) f is continuous from the right at 0.

It follows that f must be continuous on [0,∞). A modulus may be bounded or unbounded. Maddox [6] and Ruckle [9] used the modulus f to construct sequence spaces. In this paper, we introduce the strongly(V ,λ)-convergent sequences and give the relation between strongly(V ,λ)-convergence and strongly(V ,λ)-convergence with respect to a modulus.

2. Some sequence spaces

Definition2.1. Letfbe a modulus. We define the spaces, [V ,λ,f ]=

x=

xk : lim

n

1 λn

k∈In

fxk−L=0,for someL

, [V ,λ,f ]0=

x=

xk : lim

n

1 λn

k∈In

fxk=0

.

(2.1)

304 E. SAVA¸S

Whenλn=nthen the sequence spaces defined above become w0(f )and w(f ), respectively, wherew0(f )andw(f )are defined by Maddox [6].

Note that if we putf (x)=x, then we have[V ,λ,f ]=[V ,λ]and[V ,λ,f ]0=[V ,λ]0, where

[V ,λ]0=

x= xk

: lim

n

1 λn

k∈In

xk=0

. (2.2)

We have the following result.

Theorem2.2. The spaces[V ,λ,f ]and[V ,λ,f ]0are linear spaces.

Proof. We consider only[V ,λ,f ]. Suppose that xi→Land yj→L in[V ,λ,f ] and thatα,βare inC. Then there exists integersTαandMβsuch that|α| ≤Tαand

|β| ≤Mβ. We therefore have 1

λn

k∈In

fαxk+βxk−

αL+βL

≤Tα 1 λn

k∈In

fxk−L+Mβ 1 λn

k∈In

fxk−L.

(2.3)

This implies thatαx+βy→αL+βLin[V ,λ,f ]. This completes the proof.

Proposition2.3(see [7]). Letfbe any modulus. Thenlim_t→∞f (t)/t=βexists.

Proposition2.4. Letf be a modulus and let0< δ <1. Then for eachx≥δwe havef (x)≤2f (1)δ⁻¹x.

This can be proved by using the techniques similar to those used in Maddox [6] and hence we omit the proof.

Theorem2.5. Letfbe any modulus. Iflim_t→∞f (t)/t=β>0, then[V ,λ,f ]=[V ,λ]. Proof. Ifx∈[V ,λ], then

sn= 1 λn

k∈In

xk−L →0 asn → ∞,for someL. (2.4)

Letε >0 and chooseδwith 0< δ <1 such thatf (t) < εfor everytwith 0≤t≤δ.

We can write 1

λn

k∈In

fxk−L= 1 λn

k∈In,|x_k−L|≤δ

fxk−L+ 1 λn

k∈In,|x_k−L|>δ

fxk−L

≤ 1 λn

λn·ε

+2f (1)δ⁻¹sn,

(2.5)

byProposition 2.4, asn→ ∞. Thereforex∈[V ,λ,f ]. It is trivial that[V ,λ,f ]⊂[V ,λ]

and this completes the proof.

3. λ-statistical convergence. In [3], Fast introduced the idea of statistical conver- gence, which is closely related to the concept of natural density or asymptotic density of subsets of the positive integersN. In recent years, statistical convergence has been studied by several authors [1,2,4,8,10].

A sequencex=(xk)is said to be statistically convergent to the number Lif for everyε >0,

limn

1

n k≤n:xk−L≥ε=0, (3.1) where the vertical bars indicate the number of elements in the enclosed set. In this case we writes−limx=Lorxk→L(s)andsdenotes the set of all statistically convergent sequences.

In this section, we introduce and study the concept ofλ-statistical convergence and find its relation with[V ,λ,f ]andsλ.

Definition3.1. A sequencex=(xk) is said to beλ-statistically convergent or sλ-convergent toLif for everyε >0,

limn

1

λn k∈In:xk−L≥ε=0. (3.2) In this case, we write sλ−limx=L or xk→L(sλ) and sλ = {x : for someL, sλ− limx= L}. Note that ifλn=n, thensλis same ass.

The following definition was introduced by Connor [2] as an extension of the original definition of statistical convergence which appeared in [3].

Definition3.2. LetAbe a nonnegative regular summability method and letxbe a sequence. Thenxis said to beA-statistically convergent toLifχS(x−Le:ε)is contained inw0(A)for everyε >0, where

w0(A)= x: lim

n

an,kxk=0

. (3.3)

In the above definition, if we define the matrix by

an,k=





 1

λn, ifn∈In, 0, ifn∈In

(3.4)

we getλ-statistical convergence as a special case ofA-statistical convergence.

Let∇denote the set of all nondecreasing sequencesλ=(λn)of positive numbers tending to∞such thatλn+1≤λn+1 andλ1=1.

We have the following result.

Theorem3.3. Letλ∈ ∇andf be any modulus. Then[V ,λ,f ]⊂(sλ).

306 E. SAVA¸S Proof. Suppose thatε >0 andx∈[V ,λ,f ]. Since,

1 λn

k∈In

fxk−L≥ 1 λn

k∈In,|x_k−L|≥ε

fxk−L

≥ 1

λnf (ε)· k∈In:xk−L≥ε (3.5) from which it follows thatx∈(sλ). This completes the proof.

Theorem3.4. (sλ)=[V ,λ,f ]if and only iffis bounded.

Proof. Suppose thatfis bounded and thatx∈(sλ). Sincef is bounded, there is a constantMsuch thatf (x)≤Mfor allx≥0. Givenε >0, we have

1 λn

k∈In

fxk−L≤ 1 λn

k∈In,|xk−L|≥ε

fxk−L+ 1 λn

k∈In,|xk−L|<ε

fxk−L

≤ M

λn k∈In:xk−L≥ε+f (ε).

(3.6)

Taking the limit as ε→0, the result follows. Conversely, suppose that f is un- bounded so that there is a positive sequence 0< t1< t2<···< ti<··· such that f (ti)≥λi. Define the sequencex=(xi)by puttingxki=tifori=1,2,...andxi=0 otherwise. Then we havex∈(sλ), butx∈[V ,λ,f ].

References

[1] J. Connor, The statistical and strong p-Cesàro convergence of sequences, Analysis8 (1988), no. 1-2, 47–63.

[2] ,On strong matrix summability with respect to a modulus and statistical conver- gence, Canad. Math. Bull.32(1989), no. 2, 194–198.

[3] H. Fast,Sur la convergence statistique, Colloq. Math.2(1951), 241–244 (1952) (French).

[4] J. A. Fridy,On statistical convergence, Analysis5(1985), no. 4, 301–313.

[5] L. Leindler, Über die verallgemeinerte de la Vallée-Poussinsche Summierbarkeit allge- meiner Orthogonalreihen, Acta Math. Acad. Sci. Hungar. 16 (1965), 375–387 (German).

[6] I. J. Maddox,Sequence spaces defined by a modulus, Math. Proc. Cambridge Philos. Soc.

100(1986), no. 1, 161–166.

[7] ,Inclusions betweenFKspaces and Kuttner’s theorem, Math. Proc. Cambridge Phi- los. Soc.101(1987), no. 3, 523–527.

[8] D. Rath and B. C. Tripathy,On statistically convergent and statistically Cauchy sequences, Indian J. Pure Appl. Math.25(1994), no. 4, 381–386.

[9] W. H. Ruckle,FKspaces in which the sequence of coordinate vectors is bounded, Canad.

J. Math.25(1973), 973–978.

[10] T. Šalát,On statistically convergent sequences of real numbers, Math. Slovaca30(1980), no. 2, 139–150.

E. Sava¸s: Department of Mathematics, Yüzüncü Yil University, Van, Turkey E-mail address:[email protected]

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